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Question: If $\sqrt{1+y}+y \sqrt{1+x}=0$, prove that $(1+x)^{2} \frac{d y}{d x}+1=0$ Solution: We are given with an equation $x y^{2}=1$, we have to prove that $2 \frac{d y}{d x}+y^{3}=0$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove But first we need to simplify this equation in accordance with our result, which is that in our result there is no square root and our derivative is only in the form of $x$. $x \sq...
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Question: If $\sqrt{1+y}+y \sqrt{1+x}=0$, prove that $(1+x)^{2} \frac{d y}{d x}+1=0$ Solution: We are given with an equation $x y^{2}=1$, we have to prove that $2 \frac{d y}{d x}+y^{3}=0$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove But first we need to simplify this equation in accordance with our result, which is that in our result there is no square root and our derivative is only in the form of $x$. $x \sq...
Read More →A vendor purchased bananas at ₹ 40 per dozen and sold them at 10 for ₹ 36.
Question: A vendor purchased bananas at ₹ 40 per dozen and sold them at 10 for ₹ 36. Find his gain or loss per cent. Solution: LCM of 12 and $10=60$ Let the number of banana bought be $60 .$ CP of 12 banana $=₹ 40$ $\therefore$ CP of 1 banana $=\frac{40}{12}=₹ \frac{10}{3}$ $\Rightarrow$ CP of 60 bananas $=60 \times \frac{10}{3}=₹ 200$ SP of 10 bananas $=₹ 36$ $\therefore$ SP of 1 banana $=\frac{36}{10}=₹ \frac{18}{5}$ $\Rightarrow$ SP of 60 bananas $=60 \times \frac{18}{5}=₹ 216$ Here, SP of 60...
Read More →Can the following groups of elements
Question: Can the following groups of elements be classified as Dobereiners triad ? (a) Na, Si, Cl (b) Be, Mg, Ca Solution: Atomic mass of Be = 9; Na = 23; Mg = 24; Si = 28; Cl = 35; Ca = 40 Explain by giving reason. (a) No, these elements cannot be classified as triads because these do not have same properties. However, the atomic mass of Si (28) is almost the mean of the atomic masses of elements Na (23) and Cl (35). Note :The Dobereiners triad is meaningful only if the elements present in the...
Read More →Oranges are bought at 6 for ₹ 20 and sold at 4 for ₹ 18.
Question: Oranges are bought at 6 for ₹ 20 and sold at 4 for ₹ 18. Find the gain or loss per cent. Solution: LCM of 6 and $4=12$ Let the number of oranges bought be $12 .$ CP of 6 oranges $=₹ 20$ So, CP of 1 orange $=\frac{20}{6}=₹ \frac{10}{3}$ CP of 12 orange $=12 \times \frac{10}{3}=₹ 40$ SP of 4 oranges $=₹ 18$ SP of 1 orange $=\frac{18}{4}=₹ \frac{9}{2}$ SP of 12 oranges $=12 \times \frac{9}{2}=₹ 54$ Here, SP of 12 oranges $\mathrm{CP}$ of 12 oranges. Profit $=\mathrm{SP}-\mathrm{CP}=₹ 54-₹...
Read More →Elements have been arranged in the following sequence
Question: Elements have been arranged in the following sequence on the basis of their increasing atomic masses. F, Na, Mg, Al, Si, P, S, Cl, Ar, K (a) Pick two sets of elements which have similar properties. (b) The given sequence represents which law of classification of elements ? Solution: (a) The elements that have similar properties belong to the same group. From the list of elements available, elements which belong to same group are : . Na, K (Alkali metals) ; F, Cl (Halogens) (b) The sequ...
Read More →By selling 45 lemons, a vendor loses a sum equal to the selling price of 3 lemons.
Question: By selling 45 lemons, a vendor loses a sum equal to the selling price of 3 lemons. Find his loss per cent. Solution: Let Rs $x$ be the SP of one lemon. SP of 45 lemons $=$ Rs. $45 x$ Loss $=\mathrm{SP}$ of 3 lemons $=$ Rs. $3 x$ But loss $=\mathrm{CP}-\mathrm{SP}$ $\mathrm{CP}=$ loss $+\mathrm{SP}$ $=3 x+45 x$ $=$ Rs. $48 x$ $\therefore$ Loss percentage $=\left(\frac{\text { loss }}{\mathrm{CP}} \times 100\right) \%$ $=\left(\frac{3 x}{48 x} \times 100\right) \%$ $=6 \frac{1}{4} \%$...
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Question: If $x y^{2}=1$, prove that $2 \frac{d y}{d x}+y^{3}=0$ Solution: We are given with an equation $x y^{2}=1$, we have to prove that $2 \frac{d y}{d x}+y^{3}=0$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove, so by differentiating the equation on both sides with respect to $x$, we get, $y^{2}(1)+2 x y \frac{d y}{d x}=0$ $\frac{d y}{d x}=\frac{-y}{2 x}$ Or we can further solve it by using the given equatio...
Read More →The three elements A, B and C with similar properties
Question: The three elements A, B and C with similar properties have atomic masses X, Y and Z respectively. The mass of Y is approximately equal to the average mass of X and Z. What is such an arrangement of elements called as ? Give one example of such a set of elements. Solution: The arrangement is known as Dobereiners triad. For example, Calcium (Ca), Strontium (Sr) and Barium (Ba)....
Read More →By selling 130 cassettes, a man gains an amount equal to the selling price of 5 cassettes.
Question: By selling 130 cassettes, a man gains an amount equal to the selling price of 5 cassettes. Find the gain per cent. Solution: It is given that, Gain = SP of 5 cassettes .....(1) Gain = SP of 130 cassettes CP of 130 cassettes ⇒ SP of 5 cassettes = SP of 130 cassettes CP of 130 cassettes [From (1)] ⇒ CP of 130 cassettes = SP of 125 cassettes .....(2) Let the CP of 1 cassettte be ₹x. CP of 125 cassettes = ₹125x CP of 130 cassettes = ₹130x SP of 125 cassettes = CP of 130 cassettes [From (2)...
Read More →Which of the following set of elements
Question: Which of the following set of elements is written in order of their increasing metallic character ? (a) Be Mg Ca (b)Na Li K (c)Mg Al Si (d)C O N Solution: (a).These elements belong to group 2. These are written in increasing order of their size. Since the metallic character increases down the group, the order is the correct....
Read More →On moving from left to right in a period
Question: On moving from left to right in a period in the periodic table, the size of the atom (a)increases (b)decreases (c)does not change appreciably (d)first decreases and then increases Solution: (b).Atomic size decreases along a period. However, the noble gas atom is an exception. It has very large size....
Read More →The cost price of 12 candles is equal to the selling price of 15 candles.
Question: The cost price of 12 candles is equal to the selling price of 15 candles. Find the loss per cent. Solution: Let Rs $x$ be the CP of one candle and Rs. $y$ be the SP of one candle. Now, $\mathrm{CP}$ of 12 candles = SP of 15 candles $\Rightarrow 12 x=15 y$ $\Rightarrow \frac{y}{x}=\frac{12}{15}$ Loss $=\mathrm{CP}-\mathrm{SP}$ $=\mathrm{Rs}(x-y)$ $\therefore$ Loss percentage $=\left(\frac{\text { loss }}{\mathrm{CP}} \times 100\right) \%$ $=\left\{\left(\frac{x-y}{x}\right) \times 100\r...
Read More →Which one of the following does not increase
Question: Which one of the following does not increase while moving down the group of the periodic table ? (a)Atomic radius (b)Metallic character (c)Valence (d)Number of shells in an element. Solution: (c).In a group, the valence or valency does not change since all the elements present have same valence shell configuration....
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Question: If $x y=1$, prove that $\frac{d y}{d x}+y^{2}=0$ Solution: We are given with an equation $x y=1$, we have to prove that $\frac{d y}{d x}+y^{2}=0$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove, so by differentiating the equation on both sides with respect to $x$, we get, By using product rule, we get, $y(1)+x \frac{d y}{d x}=0$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y}}{\mathrm{x}}$ Or we can ...
Read More →Find the domain and the range of each of the following real function
Question: Find the domain and the range of each of the following real function: f(x) $=\frac{1}{\sqrt{2 x-3}}$ Solution: Given: $f(x)=\frac{1}{\sqrt{2 x-3}}$ Need to find: Where the functions are defined. Let, $f(x)=\frac{1}{\sqrt{2 x-3}}=y$ .......(1) The condition for the function to be defined, $2 x-30$ $\Rightarrow x\frac{3}{2}$ So, the domain of the function is the set of all the real numbers greater than $\frac{3}{2}$. The domain of the function, $\mathrm{D}_{\mathrm{f}(\mathrm{x})}=\left(...
Read More →Which one of the following depicts
Question: Which one of the following depicts the correct representation of atomic radius (r) of an atom ? (a)(i) and (ii) (b)(ii) and (iii) (c)(iii) and (iv) (d)(i) and (iv). Solution: (b).The atom (ii) has only one shell (K-shell). No electrons are present in the other shells. Therefore, the arrow represents correct atomic radius. The arrow in atom (iii) also represents the correct atomic radius of the element....
Read More →If the selling price of 16 water bottles is equal to the cost price of 17 water bottles,
Question: If the selling price of 16 water bottles is equal to the cost price of 17 water bottles, find the gain per cent earned by the dealer. Solution: Let Rs $x$ be the SP of each bottle and Rs $y$ be the CP of each bottle. SP of 16 bottles $=\mathrm{CP}$ of 17 bottles $\Rightarrow 16 \mathrm{x}=17 \mathrm{y}$ $\Rightarrow \frac{\mathrm{x}}{\mathrm{y}}=\frac{17}{16}$ Gain per bottle $=\mathrm{SP}-\mathrm{CP}$ $=\mathrm{Rs}(\mathrm{x}-\mathrm{y})$ $=\operatorname{Rs}(x-y)$ $\therefore$ Gain pe...
Read More →The element with atomic number 14 is hard and
Question: The element with atomic number 14 is hard and forms acidic oxide and a covalent halide. To which of the following categories does the element belong ? (a)Metal (b)Metalloid (c)Non-metal (d)Left-hand side element. Solution: (b).The element with atomic number (Z) = 14 is silicon (Si). It is a metalloid....
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Question: If $\mathrm{y} \sqrt{1-\mathrm{x}^{2}}+\mathrm{x} \sqrt{1-\mathrm{y}^{2}}=1$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\sqrt{\frac{1-\mathrm{y}^{2}}{1-\mathrm{x}^{2}}}$. Solution: We are given with an equation $y \sqrt{1-x^{2}}+x \sqrt{1-y^{2}}=1$, we have to prove that $\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove, so by differentiating the equation...
Read More →Which of the following elements will form an acidic oxide ?
Question: Which of the following elements will form an acidic oxide ? (a)An element with atomic number 7 (b)An element with atomic number 3 (c)An element with atomic number 12 (d)An element with atomic number 19. Solution: (a).The element with atomic number (Z) = 7 is nitrogen. It forms acidic oxides such as N2O3, N2O5etc. All other elements are metals and they form basic oxides....
Read More →Three elements B, Si and Ge are
Question: Three elements B, Si and Ge are (a)metals (b)non-metals (c)metalloids (d)metal, non-metal and metalloid respectively. Solution: (c).is the correct answer. These are also called semi-metals and possess the characteristics of both metals and non-metals....
Read More →Find the domain and the range of each of the following real
Question: Find the domain and the range of each of the following real function: $f(x)=\frac{x^{2}-16}{x-4}$ Solution: Given: $f(x)=\frac{x^{2}-16}{x-4}$ Need to find: Where the functions are defined To find the domain of the function f(x) we need to equate the denominator of the function to 0. Therefore $x-4=0$ $\Rightarrow x=4$ It means that the denominator is zero when $x=4$ So, the domain of the function is the set of all the real numbers except 4 . The domain of the function, $\mathrm{Df}(x)...
Read More →What type of oxide would
Question: What type of oxide would Eka-aluminium form ? (a)EO3 (b)E3O2 (c)E2O3 (d)EO. Solution: (c).Is the correct answer....
Read More →Coffee costing Rs 250 per kg was mixed with chicory costing Rs 75 per kg in the ratio 5 : 2 for a certain blend.
Question: Coffee costing Rs 250 per kg was mixed with chicory costing Rs 75 per kg in the ratio 5 : 2 for a certain blend. If the mixture was sold at Rs 230 per kg, find the gain or loss per cent. Solution: Let $5 \mathrm{~kg}$ of coffee be mixed with $2 \mathrm{~kg}$ of chicory. $\mathrm{CP}$ of the mixture $=\mathrm{Rs}(250 \times 5+75 \times 2)$ $=\mathrm{Rs}(1250+150)$ $=$ Rs. 1400 $\mathrm{SP}$ of the mixture $=\mathrm{Rs}(7 \times 230)=\mathrm{Rs} .1610$ Since SP $\mathrm{CP}$, there is a ...
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